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OPTIMAL SENSOR/ACTUATOR LOCATIONS FOR ACTIVE STRUCTURAL ACOUSTIC CONTROL

Sharon L. Padula and Daniel L. Palumbo NASA Langley Research Center Hampton, VA

Rex K. Kincaid The College of William and Mary Williamsburg, VA

Thirty-ninth AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference

AIAA Paper No. 98-1865 Long Beach, California April 20-23, 1998

AIAA-98-1865

OPTIMAL SENSOR/ACTUATOR LOCATIONS FOR ACTIVE STRUCTURAL ACOUSTIC CONTROL S. L. Padula∗ and D. L. Palumbo* NASA Langley Research Center, Hampton, VA R. K. Kincaid✝ The College of William and Mary, Williamsburg, VA Abstract

primary source), has a long history. In a recent 1 survey paper, Fuller and Von Flotow describe practical demonstrations of the technique as early as 1953 and a U.S. patent as early as 1936. In addition, these authors describe several commercially successful active noise and vibration control systems in use today. Their paper is highly recommended to any reader who desires a complete discussion of active acoustic control and its practical uses.

Researchers at NASA Langley Research Center have extensive experience using active structural acoustic control (ASAC) for aircraft interior noise reduction. One aspect of ASAC involves the selection of optimum locations for microphone sensors and force actuators. This paper explains the importance of sensor/actuator selection, reviews optimization techniques, and summarizes experimental and numerical results.

The scope of the present paper is limited to active structural acoustic control (ASAC), with a focus on aircraft interior noise control research conducted at NASA Langley Research Center. The most obvious difference between the ASAC system and early acoustic control systems is that ASAC uses structural actuators like shakers or piezoelectric (PZT) patches attached to the aircraft fuselage rather than acoustic actuators like loudspeakers inside the fuselage. The ASAC concept is attractive because the structural actuators are more effective by weight and consume less interior volume than competing active or passive 2 noise control options.

Three combinatorial optimization problems are described. Two involve the determination of the number and position of piezoelectric actuators, and the other involves the determination of the number and location of the sensors. For each case, a solution method is suggested, and typical results are examined. The first case, a simplified problem with simulated data, is used to illustrate the method. The second and third cases are more representative of the potential of the method and use measured data. The three case studies and laboratory test results establish the usefulness of the numerical methods.

One area of ASAC research is the determination of optimal locations for actuators and sensors. Early 3−5 theoretical investigations established the importance of actuator and sensor architecture and suggested

Introduction Active acoustic control, or the use of one acoustic source (or secondary source) to cancel another (or ∗

Engineer, Fluid Mechanics and Acoustics Division, Senior Member AIAA Professor, Department of Mathematics Copyright  1998 by the American Institute of Aeronautics and Astronautics, Inc. No copyright is asserted in the United States under Title 17, U.S. Code. The U.S. Government has a royalty-free license to exercise all rights under the copyright claimed herein for Governmental Purposes. All other rights are reserved by the copyright owner. ✝

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optimization strategies and goals. For example, 3 Silcox et al. introduce a mathematical model for ASAC and demonstrate that for fixed actuator and sensor locations the force inputs that yield minimum interior noise are easy to calculate. Ruckman and 4 Fuller suggest that the best actuator locations can be found by selecting subsets of actuators from a large set of candidates. They further suggest statistical measures that test whether seemingly good actuator sets will perform well in spite of measurement noise and numerical errors. In reference 5, Padula and Kincaid solve the ASAC actuator subset selection problem by using a combinatorial search method that was originally applied to spacecraft optimization problems. References 6 and 7 supply details in regard to the search method and suggest several modifications that improve its usefulness and efficiency.

Three combinatorial optimization problems are described in this report. All involve determination of the best positions for ASAC sensors and actuators. For each case, a solution method is suggested, and typical results are examined. The first case is a simplified problem with simulated data that is used to illustrate the method. The second case applies the technique to the laboratory model used by Lyle and Silcox and compares the automated procedure with the modal method. The final case is more representative of the potential of the method and shows how the method can be extended to include a large number of actuators that must reduce noise at multiple frequencies and that have realistic force limits. Optimization Overview

1

In this section, a combinatorial optimization method for selecting actuator (or sensor) locations is described. All three test cases in this paper use some variation of this generalized algorithm.

In their survey paper, Fuller and Von Flotow describe the actuator location problem from a practical standpoint. They note that researchers recommend a modal method, such that actuators are placed to excite a selected structural mode and sensors are placed to observe each important acoustic 2 mode. Lyle and Silcox tested this modal method on a simulated aircraft fuselage with mixed results. They demonstrated impressive global interior noise reduction at a frequency at which both the primary and secondary sources excited the same dominant acoustic mode. However, at a second frequency, the same actuator and sensor configuration should have been effective, yet when tested a global increase in interior noise was noted. Lyle and Silcox explain that in the second case several acoustic modes were important and, although the controller successfully reduced the dominant mode, several other modes were amplified (i.e., control spillover was observed). Further, the authors demonstrated that an alternate set of actuators and sensors greatly reduced the spillover effect.

Given a set of Na actuator locations, the goal of an optimization run is to identify a subset of Nc locations that provides the best performance (e.g., reduces interior noise). Several combinatorial optimization methods, such as simulated annealing, genetic algorithms, and tabu search, are available. Tabu search was selected for use in the present study, based 5 on previous experience. To apply a tabu search algorithm one must define a state space, a method for moving from state to state, a neighborhood for each state, and a cost function to minimize. For the actuator selection problem, the set of all possible subsets of size Nc chosen from Na actuators is the state space. To bound the problem, the subset size Nc is constant for each search. An initial state can be prescribed by the user or can be generated randomly. At any given state, the subset Nc, of actuators that represent that state are flagged as "on;" the remaining actuators are flagged as "off." A move changes the state by turning one actuator off and one on. A neighborhood is the set of all states that are one move away from the current state. Finally, the cost function is based on the noise reduction estimate for the subset of actuators that are turned on.

This report reviews combinatorial optimization techniques for actuator and sensor location problems. The report extends the material in reference 5 in several ways. First, the optimization techniques are demonstrated by application to test articles. Next, the optimized sensor and actuator architectures are compared with those derived by modal methods. Finally, the optimization methods are extended to cases in which several frequencies are controlled simultaneously and in which actuator excitation voltages cannot exceed transducer saturation limits.

Each iteration of the tabu search algorithm involves evaluating the cost function for each subset of

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predictable pressure waves that are exterior to the cylinder. These periodic pressure changes cause predictable structural vibrations in the cylinder wall and predictable noise levels in the interior space. The interior noise level at any discrete microphone location can be dramatically reduced by using the PZT actuators to modify the vibration of the cylinder. For a given set of microphones and a given set of actuator locations, the control forces that minimize 3 the acoustic response are known. However, methods for choosing good locations for the microphones and 8 the actuators are needed. In accordance with the notation used in reference 9, the ASAC optimization problem is to minimize the sum of squared pressures at a set of Np interior microphones:

actuators in the neighborhood of the current state. The move that improves the cost function the most is accepted. If no improving move is identified, then the move that degrades the cost function the least is accepted. The algorithm continues for a predetermined number of iterations. Cycling is avoided by maintaining a list (called the tabu list) of all accepted moves. The algorithm is prohibited from reversing any move on the tabu list. An exception can be made if the move is old or if the move produces a state that is clearly better than any previous state. The algorithm terminates after reporting the best state that was encountered during the entire optimization procedure. Note that each iteration of tabu search requires Nc*(Na − Nc) evaluations of the cost function. For example, in the first test case, tabu search is used to select 16 actuator locations from a possible 102. This scenario requires 16*86 = 1376 evaluations per iteration. Typical searches require at least 15 5 iterations, or approximately 2 x 10 evaluations. This number of evaluations is small in comparison with the total number of possible actuator combinations 18 (~2 x 10 ) but can be significant if the cost function is computationally expensive. Selection of the least computationally expensive cost function that maintains the relative ranking of the actuator sets in the search space is desirable. The cost function is not required to be smooth or continuous or quantitatively accurate. The only requirement is that the cost function can identify the better of two actuator sets.

Np

E = ∑ Λ m Λ*m

(1)

m =1

where * indicates the complex conjugate. response at microphone m is given as

The

Nc

Λ m = ∑ Hmk c k + p m

(2)

k=1

where pm is the response with no active control and Hmk is a complex-valued transfer matrix that represents the response at microphone m that results from one unit control force (|ck| = 1) at actuator k. The values in the transfer matrix can be collected experimentally (ref. 2), or they can be simulated (ref. 3). The cost function can be written either as in equation (1) or on a decibel scale to compare the interior pressure norms with and without ASAC:

Simulated ASAC

 Np  * ΛmΛm ∑   (3) Level = 10 log m =1 Np  ∑ pm p*m   m =1

In this section, tabu search is applied to a simplified model of the ASAC problem. This simulation serves several purposes. First, the simulation illustrates the method and indicates the potential for reducing aircraft interior noise. Secondly, the relationship between the shell vibration level and the interior noise level is explored.

A negative level represents a decrease in sound pressure level caused by the action of PZT actuators.

Problem Statement

For a fixed set of Nc actuators, the forces ck that minimize either equation (1) or equation (3) can be determined by solving a complex least-squares 3 problem. Unfortunately, the solution vector may contain values of ck that exceed the maximum allowable control force. Also, for some sets of actuators the solution vector decreases the interior noise level but increases the shell vibration level. (Note that an equation similar to equation (3) exists that compares the vibration norms with and without

Assume that an aircraft fuselage is represented as a cylinder with rigid end caps (fig. 1) and that a propeller is represented as a point monopole with a frequency equal to some multiple of the blade passage frequency. Piezoelectric actuators bonded to the fuselage skin are represented as line force distributions in the x and θ directions. With this simplified model, the point monopole produces

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of actuators up or down from 16, the noise-reduction goals can be satisfied without an increase in vibration and without exceeding the force capacity of the PZT actuators.

ASAC. A positive vibration level indicates an increase in shell vibration as a result of the action of PZT actuators.) Effective noise control strategies can either reduce the vibration of the cylinder or magnify the vibration of the cylinder by shifting vibrational energy to shell modes that do not couple efficiently 8 This insight is important with acoustic modes. because aircraft manufacturers may reject a noise control method that increases vibration and, in turn, increases the potential for fatigue failure of the airframe.

The best locations for PZT actuators are not intuitively obvious. For example, figure 3 shows the grid of 102 possible locations distributed in 6 rings of 17 locations each. Each actuator location is specified by the (x, θ, r = a) position of its center. (Recall fig. 1.) The acoustic monopole is located at (x = L/2, θ = 0, r = 1.2a), where L is the cylinder length and a is the cylinder radius. (The dimensions of the cylinder are typical of commuter aircraft configurations, and the frequency of the source simulates harmonics of typical turboprop blade passage frequencies.) The shaded rectangles indicate the 16 best actuator locations. Figure 3(a) shows the best locations for controlling interior noise caused by an acoustic monopole with a frequency of 200 Hz. Figure 3(b) indicates the change in the best locations for an acoustic monopole with a frequency of 275 Hz. Notice the symmetric pattern in figure 3(a) that corresponds to a case in which the acoustic monopole excites one dominant interior cavity mode. Notice the greater complexity of the pattern in figure 3(b). Here, several cavity modes of similar importance are excited by the monopole with a frequency of 275 Hz.

One approach to this problem of minimizing noise and vibration is to assume that the forces ck are variable but their locations are fixed. For example, reference 9 uses a multiobjective optimization formulation to trade off noise reduction and vibration reduction while imposing force constraints. This formulation can be successful but is highly sensitive to the weights placed on each objective. An alternate way to pose the problem is to make the control forces dependent variables and choose the number and locations of the actuators. Given a large number Na of possible locations and a transfer matrix a H that includes the response for each possible actuator, the alternate procedure uses tabu search to converge to the best Nc